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Blumberg theorem – even if a real function : is nowhere continuous, there is a dense subset of such that the restriction of to is continuous. Thomae's function (also known as the popcorn function) – a function that is continuous at all irrational numbers and discontinuous at all rational numbers.
Discontinuous functions may be discontinuous in a restricted way, giving rise to the concept of directional continuity (or right and left continuous functions) and semi-continuity. Roughly speaking, a function is right-continuous if no jump occurs when the limit point is approached from the right.
A natural follow-up question one might ask is if there is a function which is continuous on the rational numbers and discontinuous on the irrational numbers. This turns out to be impossible. The set of discontinuities of any function must be an F σ set. If such a function existed, then the irrationals would be an F σ set.
However, not all functions are continuous. If a function is not continuous at a limit point (also called "accumulation point" or "cluster point") of its domain, one says that it has a discontinuity there. The set of all points of discontinuity of a function may be a discrete set, a dense set, or even the entire domain of the function.
When we speak of a function being continuous on an interval, we mean that the function is continuous at every point of the interval. In contrast, uniform continuity is a global property of f {\displaystyle f} , in the sense that the standard definition of uniform continuity refers to every point of X {\displaystyle X} .
Then f is a non-decreasing function on [a, b], which is continuous except for jump discontinuities at x n for n ≥ 1. In the case of finitely many jump discontinuities, f is a step function . The examples above are generalised step functions; they are very special cases of what are called jump functions or saltus-functions.
The pointwise limit of continuous functions does not have to be continuous: the continuous functions (marked in green) converge pointwise to a discontinuous function (marked in red). Suppose that X {\displaystyle X} is a set and Y {\displaystyle Y} is a topological space , such as the real or complex numbers or a metric space , for example.
A continuous function () on the closed interval [,] showing the absolute max (red) and the absolute min (blue). In calculus , the extreme value theorem states that if a real-valued function f {\displaystyle f} is continuous on the closed and bounded interval [ a , b ] {\displaystyle [a,b]} , then f {\displaystyle f} must attain a maximum and a ...